Analytic Function

A function $f$ is analytic at $z_{0}$ if there exists $R > 0$ and coefficients $a_{n}$ such that for all $z$ with $| z - z_{0} | < R$ :

f (z) = \sum_{n = 0}^{\infty} a_{n} (z - z_{0})^{n}

Radius $R$ is the radius of convergence of the series.

It is a theorem that every power series converges uniformly on compact subsets of its disk of convergence.
Complex case: $f$ analytic on open set $U$ if analytic at each $z_{0} \in U$ . It is equivalent to being a holomorphic function at $z_{0}$ . See Taylor's theorem.
Real case: same definition for $f : R \to R$

On the other hand, any power series has a radius, $\bar{R}$ , such that

converges for $| z - z_{0} | < \bar{R}$
diverges for $| z - z_{0} | > \bar{R}$
It is a theorem that $\bar{R}$ equals the distance from $z_{0}$ to the nearest singularity of the analytic continuation of the function $f (z) = \sum_{n = 0}^{\infty} a_{n} (z - z_{0})^{n}$ .